Average Error: 14.3 → 0.4
Time: 4.1m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{-2}{(x \cdot \left(x + -1\right) + \left(x + -1\right))_*}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{-2}{(x \cdot \left(x + -1\right) + \left(x + -1\right))_*}
double f(double x) {
        double r33019596 = 1.0;
        double r33019597 = x;
        double r33019598 = r33019597 + r33019596;
        double r33019599 = r33019596 / r33019598;
        double r33019600 = r33019597 - r33019596;
        double r33019601 = r33019596 / r33019600;
        double r33019602 = r33019599 - r33019601;
        return r33019602;
}
↓
double f(double x) {
        double r33019603 = -2.0;
        double r33019604 = x;
        double r33019605 = -1.0;
        double r33019606 = r33019604 + r33019605;
        double r33019607 = fma(r33019604, r33019606, r33019606);
        double r33019608 = r33019603 / r33019607;
        return r33019608;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.3

    \[\frac{1}{x + 1} - \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied frac-sub13.7

    \[\leadsto \color{blue}{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot \left(x - 1\right)}}\]

  4. Simplified0.4

    \[\leadsto \frac{\color{blue}{-2}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]

  5. Simplified0.4

    \[\leadsto \frac{-2}{\color{blue}{(x \cdot \left(x + -1\right) + \left(x + -1\right))_*}}\]

  6. Final simplification0.4

    \[\leadsto \frac{-2}{(x \cdot \left(x + -1\right) + \left(x + -1\right))_*}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))